If we are considering functions defined on the whole of ℝm or an unbounded subset of ℝm, the Cn norm may be infinite. For example,

∥ex∥Cn=∞

for all n because the n-th derivative of ex is again ex, which blows up as x approaches infinity. If we are considering functions on a compact (closed and bounded) subset of ℝm however, the Cn norm is always finite as a consequence of the fact that every continuous function on a compact set attains a maximum. This also means that we may replace the “sup” with a “max” in our definition in this case.

Having a sequence of functions converge under this norm is the same as having their n-th derivatives converge uniformly. Therefore, it follows from the fact that the uniform limit of continuous functions is continuous that Cn is complete under this norm. (In other words, it is a Banach space.)

In the case of C∞, there is no natural way to impose a norm, so instead one uses all the Cn norms to define the topology in C∞. One does this by declaring that a subset of C∞ is closed if it is closed in all the Cn norms. A space like this whose topology is defined by an infinite collection of norms is known as a multi-normed space.